We focus on dropout techniques for asynchronous distributed computations in federated learning (FL) scenarios. We propose \texttt{AsyncDrop}, a novel asynchronous FL framework with smart (i.e., informed/structured) dropout that achieves better performance compared to state of the art asynchronous methodologies, while resulting in less communication and training time costs. The key idea revolves around sub-models out of the global model, that take into account the device heterogeneity. We conjecture that such an approach can be theoretically justified. We implement our approach and compare it against other asynchronous baseline methods, by adapting current synchronous FL algorithms to asynchronous scenarios. Empirically, \texttt{AsyncDrop} significantly reduces the communication cost and training time, while improving the final test accuracy in non-i.i.d. scenarios.